Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15278 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,15,8)(4,9,19,27)(5,12)(6,10)(7,11)(14,22)(16,23,17,24)(18,32)(20,26,21,28)(25,31)(29,30);(1,3)(2,8)(4,18)(5,17)(6,16)(7,15)(9,25)(10,24)(11,13)(12,23)(14,20)(19,31)(21,30)(22,28)(26,29)(27,32);(1,4)(2,9,11,27)(3,14)(5,21)(6,20)(7,19)(8,22,13,29)(10,26,12,28)(15,30)(16,31)(17,18)(23,32,24,25);(2,10,11,12)(8,23,13,24)(9,26,27,28)(22,32,29,25)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72219482, #72219485, … |
| maximal normal subgroup of | #72215974, #72219482, #72219485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327713);